Problem 58 - Introductory Algebra for College Students

Question

Solve each equation and check your proposed solution in Exercises. $$0.05(7 x+36)=0.4 x+1.2$$

Step-by-Step Solution

Answer

The solution for the equation is \(x = -12\)

Step 1: Simplify both sides

To simplify the equation, start by distributing the 0.05 on the left side of the equation. Therefore, the equation becomes: \(0.35x + 1.8 = 0.4x + 1.2\)

Step 2: Collect like terms on each side

Subtract 0.35x from both sides of the equation to bring all the 'x' terms on one side, and subtract 1.2 from both sides to bring all the constant terms on the other side. So, the equation narrows down to: \(0.05x = -0.6\)

Step 3: Solve for x

To solve for x, divide both sides of the equation by 0.05. The equation becomes: \(x = -12\)

Step 4: Check the solution

Substitute x = -12 in the original equation to check the solution. Therefore, \(0.05(7 × -12 + 36) = -4.2\), and \(0.4 × -12 + 1.2 = -4.2\). Since both sides of the equation give the same result (-4.2), the solution is correct.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that helps simplify equations. It allows you to multiply a term across a sum or difference inside parentheses. For example, given an expression like \( a(b + c) \), the distributive property lets you multiply \( a \) by each term inside the parentheses: \( ab + ac \).
This property is especially handy when dealing with linear equations, as it helps to remove parentheses and simplify expressions.
In the exercise, you see the distributive property in action with \( 0.05(7x + 36) \). Here, the \( 0.05 \) is distributed, giving \( 0.05 \times 7x + 0.05 \times 36 \) which simplifies to \( 0.35x + 1.8 \).

• Always distribute multiplication over addition or subtraction.
• Apply this property to simplify expressions in equations before trying to solve them.

By mastering the distributive property, you can more easily manage and solve linear equations, leading to faster and more accurate solutions.

Linear Equations

Linear equations are equations where the highest power of the variable, typically denoted as \( x \), is 1. These equations form a straight line when graphed on a coordinate plane.
The form is usually expressed as \( ax + b = c \). Solutions to linear equations are values of \( x \) that make the equation true.
In our exercise, you're dealing with linear terms like \( 0.35x + 1.8 \) and \( 0.4x + 1.2 \). These are both linear expressions, and solving the equation involves manipulations to isolate \( x \).

• Begin by combining like terms.
• Use properties like addition, subtraction, multiplication, or division to isolate the variable.
• Verify the solution by substituting it back into the original equation.

Understanding and solving linear equations is a crucial skill, as it provides the basis for more complex algebraic operations.

Problem-Solving Steps

Approaching algebra problems methodically ensures you arrive at the correct solution. There are key problem-solving steps to keep in mind:

Simplify Both Sides of the Equation

Begin by using properties such as the distributive property to simplify each side of the equation. This involves distributing any terms and combining like terms.

Collect Like Terms

Next, adjust the equation so all variable terms are on one side and constants on the other. This might involve adding or subtracting terms from both sides.

Solve for the Variable

Once simplified, solve for the variable by performing operations that isolate it, such as division or multiplication.
In the exercise, this involves dividing by 0.05 to find \( x = -12 \).

Check Your Solution

Verify the solution by substituting it back into the original equation. If both sides equal, your solution is correct. In our case, substituting \( x = -12 \) confirmed the result since both sides gave \(-4.2\).
Following these steps not only aids in solving linear equations but builds confidence in handling various algebraic challenges.